We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation that is known to be finite and convergent. Finally, we show how to reduce this coherent presentation to a Tietze equivalent one having Knuth's generators. M.S.C. 2010 -20M05, 18D05, 68Q42, 05E10.
Syzygies of Knuth's relations.The aim of this work is to give an algorithmic method for the syzygy problem of finding all independent irreducible algebraic relations amongst the Knuth relations and some other presentations of the plactic monoids in type A. A 2-syzygy for a presentation of a monoid is a relation amongst relations. For instance, using the Knuth relations there are two ways to prove the equality 2211 = 2121 in the monoid P 2 , either by applying the first Knuth relation 211 = 121 or the second relation 221 = 212. This two equalities are related by a syzygy. Starting with a monoid presentation, we would like to compute all syzygies for this presentation and in particular to compute a family of generators for the syzygies. For instance, we will prove that in rank 2 the two Knuth relations form a unique generating syzzygy for the Knuth relations. For rank greater than 3, the syzygies problem for the Knuth presentation is difficult due to the combinatorial complexity of the relations. In commutative algebra, the theory of Gröbner bases gives algorithms to compute bases for linear syzygies. By a similar method, the syzygy problem for presentation of monoids can be algorithmically solved using convergent rewriting systems.Rewriting and plactic monoids. Study presentations from a rewriting approach consists in the orientation of the relations, then called reduction rules. For instance, the relations of the monoid P 2 can be oriented with respect to the lexicographic order as follows η 1,1,2 : 211 ⇒ 121 ε 1,2,2 : 221 ⇒ 212.In a monoid presented by a rewriting system, two words are equal if they are related by a zig-zag sequence of applications of reductions rules. A rewriting system is convergent if the reduction relation induced by the rules is well-founded and it satisfies the confluence property. This means that any reductions starting on a same word can be extended to end on a same reduced word. Recently plactic monoids were investigated by rewriting methods [22,3,5,14,4].
IntroductionCoherent presentations. In this paper, we give a categorical description of 2-syzygies of p...